Question

In heat capacity how could we get the specific heat capacity relation at constant volume please explain me in a detailed way

Answer 1

Heat Capacity at Constant Volume

Q = nC_VΔT

For an ideal gas, applying the First Law of Thermodynamics tells us that heat is also equal to:

Q = ΔE_int + W, although W = 0 at constant volume.

For a monatomic ideal gas ΔE_int = (3/2)nRΔT

Comparing our two equations

Q = nC_VΔT and Q = (3/2)nRΔT

we see that, for a monatomic ideal gas:

C_V = (3/2)R

For diatomic and polyatomic ideal gases we get:

diatomic: C_V = (5/2)R

polyatomic: C_V = 3R

This is from the extra 2 or 3 contributions to the internal energy from rotations.